Splits with forbidden subgraphs

TL;DR

The paper determines the minimal split parameter $f(n,H)$ for avoiding a fixed subgraph $H$ in an $(n,k)$-graph, connecting it to extremal functions ${ m ex}(ullet,H)$ and Ramsey numbers. Using a combination of probabilistic colorings, near-extremal $H$-free graphs, and explicit constructions (notably a $C_4$-free affine-plane-based graph), it establishes sharp or near-sharp bounds for broad classes of $H$: $f(n,H)=2$ for non-bipartite $H$, and $f(n,H)$ scaling as $n^{2/r-1}$ up to a logarithmic factor when ${ m ex}(ullet,H)= olinebreak \Theta(ullet^{r})$ for bipartite $H$ with Turán exponent $r$. It also provides concrete results like $f(n,K_{2,t})= olinebreak \\Theta(n^{1/3})$, a $C_4$-free construction achieving $f(n,C_4)=(1+o(1))\Theta(n^{1/3})$, and tree-specific bounds that vary by the structure of the tree. Overall, the work deepens the link between split-graph constructions and extremal graph theory, offering both general asymptotics and explicit extremal examples with potential minor-minor and representation-theoretic implications.

Abstract

In this note, we fix a graph $H$ and ask into how many vertices can each vertex of a clique of size $n$ can be "split" such that the resulting graph is $H$-free. Formally: A graph is an $(n,k)$-graph if its vertex sets is a pairwise disjoint union of $n$ parts of size at most $k$ each such that there is an edge between any two distinct parts. Let $$ f(n,H) = \min \{k \in \mathbb N : \mbox{there is an $(n,k)$-graph $G$ such that $H\not\subseteq G$}\} . $$ Barbanera and Ueckerdt observed that $f(n, H)=2$ for any graph $H$ that is not bipartite. If a graph $H$ is bipartite and has a well-defined Turán exponent, i.e., ${\rm ex}(n, H) = Θ(n^r)$ for some $r$, we show that $Ω(n^{2/r -1}) = f(n, H) = O (n^{2/r-1} \log ^{1/r} n)$. We extend this result to all bipartite graphs for which an upper and a lower Turán exponents do not differ by much. In addition, we prove that $f(n, K_{2,t}) =Θ(n^{1/3})$ for any fixed $t$.

Theorems & Definitions (19)

• Proposition 1: Barbanera, Ueckerdt Ba
• proof
• Theorem 2
• Corollary 3
• Theorem 4
• Theorem 5
• Lemma 6
• proof
• Lemma 7
• proof